A Finite Presentation of CNOT-Dihedral Operators

31 Dec 2016  ·  Matthew Amy, Jianxin Chen, Neil J. Ross ·

We give a finite presentation by generators and relations of the unitary operators expressible over the {CNOT, T, X} gate set, also known as CNOT-dihedral operators. To this end, we introduce a notion of normal form for CNOT-dihedral circuits and prove that every CNOT-dihedral operator admits a unique normal form. Moreover, we show that in the presence of certain structural rules only finitely many circuit identities are required to reduce an arbitrary CNOT-dihedral circuit to its normal form. By appropriately restricting our relations, we obtain a finite presentation of unitary operators expressible over the {CNOT, T} gate set as a corollary.

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Quantum Physics Emerging Technologies Logic in Computer Science